Lines and Their Equations: Slope, Forms, and Relationships

Chapter 1: Graphs

Section 1.3: Lines

This section introduces the concept of lines in the coordinate plane, focusing on the slope, different forms of linear equations, and the relationships between lines such as parallelism and perpendicularity. Understanding these foundational ideas is essential for further study in precalculus and calculus.

Understanding Slope

The slope of a line is a measure of its steepness, defined as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line.

• Definition: If $(x_1, y_1)$ and $(x_2, y_2)$ are two distinct points on a nonvertical line, the slope $m$ is given by: $m = \frac{y_2 - y_1}{x_2 - x_1}$

• Interpretation: The slope tells us how much $y$ changes for a unit change in $x$.

• Vertical lines: The slope is undefined because the run is zero, resulting in division by zero.

• Horizontal lines: The slope is zero because the rise is zero.

Example: Comparing Steepness

Consider two lines, a and b, with the same run but different rises. The line with the greater rise is steeper. If the rise stays the same and the run decreases, the line becomes steeper; if the run increases, the line becomes less steep.

Key Properties of Slope

• Positive slope: Line rises from left to right.

• Negative slope: Line falls from left to right.

• Zero slope: Line is horizontal.

• Undefined slope: Line is vertical.

• The greater the absolute value of the slope, the steeper the line.

Graphing Lines Given a Point and the Slope

To graph a line when given a point $(x_1, y_1)$ and a slope $m$:

• Start at the given point.

• Use the slope to determine the rise and run, then plot a second point.

• Draw the line through both points.

Equations of Lines

Vertical Lines

• The equation of a vertical line is $x = a$, where $a$ is the x-intercept.

• The slope is undefined.

Horizontal Lines

• The equation of a horizontal line is $y = b$, where $b$ is the y-intercept.

• The slope is zero.

Point-Slope Form

• If a line has slope $m$ and passes through $(x_1, y_1)$, its equation is: $y - y_1 = m(x - x_1)$

• This form is useful when you know a point and the slope.

Slope-Intercept Form

• If a line has slope $m$ and y-intercept $b$, its equation is: $y = mx + b$

• This form is useful for quickly identifying the slope and y-intercept.

General (Standard) Form

• The general form of a line is $Ax + By = C$, where $A$, $B$, and $C$ are real numbers and $A$ and $B$ are not both zero.

• Any linear equation can be written in this form.

Finding the Equation of a Line Given Two Points

1. Find the slope $m$ using the two points.

2. Use the point-slope form to write the equation.

3. Simplify to slope-intercept or general form as needed.

Graphing Lines Using Intercepts

To graph a line given in general form, find the x-intercept (set $y=0$) and y-intercept (set $x=0$), plot both points, and draw the line through them.

Parallel and Perpendicular Lines

Parallel Lines

• Two nonvertical lines are parallel if and only if they have the same slope and different y-intercepts.

• Vertical lines are parallel if they have different x-intercepts.

Perpendicular Lines

• Two nonvertical lines are perpendicular if and only if the product of their slopes is $-1$ (i.e., their slopes are negative reciprocals).

• Any vertical line is perpendicular to any horizontal line.

Example: Perpendicular Lines in the Coordinate Plane

Consider the lines $x + 3y = -9$ and $3x - y = 4$. Their graphs intersect at a right angle, confirming they are perpendicular.

Effect of Viewing Window on Perpendicularity

When graphing with technology, the appearance of right angles can be distorted if the axes are not equally scaled. A square viewing window preserves the true appearance of perpendicularity.

Summary Table: Forms of Linear Equations

Form	Equation	Key Features
Slope-Intercept	$y = mx + b$	Slope $m$, y-intercept $b$
Point-Slope	$y - y_1 = m(x - x_1)$	Point $(x_1, y_1)$, slope $m$
General (Standard)	$Ax + By = C$	Intercepts, can represent all lines
Vertical	$x = a$	Vertical line, undefined slope
Horizontal	$y = b$	Horizontal line, zero slope

Additional info: The above notes synthesize and expand upon the provided content, filling in missing definitions, formulas, and examples for completeness and clarity.